Regularity for eigenvalue-type equations in sets with thick complement

We prove global higher integrability of the gradient for solutions to nonlinear PDEs with homogeneous Dirichlet condition. We work with general open sets (i.e. not necessarily bounded), under a minimal thickness assumption of the complement. The main focus is on improving the global summability of the gradient, by preserving the ``zero trace" Sobolev class. We pay due attention to providing precise a priori estimates. As an application, we get universal global H\"older estimates for solutions of the super-homogeneous Lane-Emden equation for the $N-$Laplacian, in contractible open sets with finite inradius.